Let \(A\) and \(B\) be subsets of a topological space \(X\). Suppose that \(p\) is a limit point of the subset \(A\cup B\). Show that \(p\) is also a limit point of \(A\) or \(B\).
Solution
Recall that \(p\) is a limit point of \(C\subseteq X\) when every neighborhood of \(p\) meets \(C\setminus\{p\}\).
1. If \(p\) is a limit point of neither \(A\) nor \(B\), then some neighborhood \(W\) of \(p\) satisfies \((W\setminus\{p\})\cap(A\cup B)=\varnothing\).
There are neighborhoods \(U\), \(V\) of \(p\) with \((U\setminus\{p\})\cap A=\varnothing\) and \((V\setminus\{p\})\cap B=\varnothing\). \(W=U\cap V\) is a neighborhood of \(p\), and \(W\setminus\{p\}\) lies in both \(U\setminus\{p\}\) and \(V\setminus\{p\}\), so it misses \(A\) and \(B\).
2. Q.E.D.
Since \(p\) is a limit point of \(A\cup B\), no neighborhood \(W\) as in step 1 exists, so \(p\) is a limit point of \(A\) or of \(B\).